\documentclass[12pt]{report}
\usepackage{amsmath,amsfonts,amssymb}
\setlength{\textwidth}{18cm} \setlength{\textheight}{21cm}
\setlength{\topmargin}{-1cm} \setlength{\oddsidemargin}{-1cm}
\setlength{\evensidemargin}{-1cm}

\newcommand{\bq}{\begin{eqnarray*}}
\newcommand{\eq}{\end{eqnarray*}}
\begin{document}
\pagestyle{empty}
\begin{center}
\Large
{\bf STAT-324}\\
{\bf Midterm I}\\
\large
{\bf February 22, 2005\\
9:30-10:40AM}\\
\normalsize
Instructor: Moo K. Chung\\
\vspace{2cm}
Answer all questions showing all your work {\bf neatly}.\\
Calculators are allowed. One page note is permitted \vspace{2cm}

\large
{\bf Name:}\underline{\hspace{10.5cm}}\\

{\bf Student ID:}\underline{\hspace{9cm}}\\

\normalsize
\vspace{1cm}
Pledge: On my honor, I have neither given nor received\\
unauthorized aid on this examination.\\
\vspace{1cm}
\large
{\bf Signature:}\underline{\hspace{9cm}}\\
\vspace{2cm}

\begin{tabular}{|r|l|c|}\hline
Question & Points}\\
\hline
1 & \\
\hline
2 & \\
\hline
3 & \\
\hline
4 & \\
\hline
Total & \\
\hline
\end{tabular}
\end{center}

\newpage

\begin{enumerate}

\item (a) Suppose that $A$ and $B$ are independent events such that the probability that neither occurs is $\frac{1}{2}$ and the probability of $B$ is $\frac{1}{3}$. Determine the probability of $A$ occurring [5pts].
\vspace{12cm}

(b) If $A$ and $B$ are independent events, show that $A^c$ and $B$
are also independent [5pts].

\vspace{6cm}

\newpage


\item A lie detector will show a positive reading (indicates a lie)
$10\%$ of the time when a person is telling the truth and $90\%$ of
the time when the person is lying. Suppose two people are suspects
in a crime and only one is guilty. The guilty one will be lying
while the innocent one will be telling the truth.\\ (a) What is the
probability that the detector shows positive readings for both
suspects [5pts]?

\vspace{12cm}

(b) What is the probability that a person is lying when the detector
shows a negative reading [5pts]?
\newpage

\item A man with $n$ keys wants to open his door and tries the keys at random.
Exactly one key will open the door.\\ (a) Suppose unsuccessful keys
are {\bf not eliminated} from $n$ keys. Determine the probability
distribution of the number of trials until he opens the door [5pts].

\vspace{11cm} (b) Suppose unsuccessful keys are {\bf eliminated}
from $n$ keys. Determine the probability distribution of the number
of trials until he opens the door [5pts].

\newpage

\item We have a probability density $f(x)= kx^2$ for $0 < x < 1$ and
$f(x)=0$ otherwise.\\
(a) Compute the variance and the expected value of the distribution
[5pts].

\vspace{13cm}

(b) Determine the $75$-th percentile point [5pts]. \\
\end{enumerate}
\end{document}